Has this tetration been considered before in the forum?

[JmsNxn]

I hope you all can forgive my naivety but I found this on MathSE. It somehow seems familiar but at the same time, since I missed a lot of the nitty gritty details of the tetration extensions in the past discussions (not the beta method ones), it somehow look new to me.

In this question How to evaluate fractional tetrations? (March 2020) the user Simply Beautiful Art cites a chain of previous question of his/her, and in that the author claim the method is probably equivalent to Kneser. The formula is derived from the assumption of asymptotic behavior analogous to that of Gamma function.

The chain of questions where this is laid out are:
Dec 26, 2019 Numerical instability of an extended tetration
Dec 26, 2019 Verifying tetration properties
Dec 29, 2019 Verifying uniqueness of my tetration

Of the three only the first received attention by forum users (Gottfried). There only the question if computation was unstable of the formula non-convergent was treated. The other posts maybe were not noticed by the forum experts. So my question: is it something new? Was already discussed here?

You can prove it's the standard Schroder iteration pretty simply.

First of all, it converges for \(\Re(x) > X\) for large enough \(X\). Then notice that it is periodic with the same period as Schroder's iteration; which is \(2 \pi i / \log \log (^\infty a)\). Then notice that it's bounded in the right half plane \(\Re(x) > X\). Then notice that it interpolates the values \(^na\) for \(n > X\). Now pull out your Ramanujan identity theorem:

If two functions \(F,G\) are holomorphic and bounded for \(\Re(s) > X\) and interpolate the same values, \(F = G\).

Easy peasy.

I have never in my life seen that expression for tetration though. Especially odd looking...